Optimal extensions for p -th power factorable operators
aa r X i v : . [ m a t h . F A ] N ov OPTIMAL EXTENSIONS FOR p -TH POWER FACTORABLEOPERATORS O. DELGADO AND E. A. S ´ANCHEZ P´EREZ
Abstract.
Let X ( µ ) be a function space related to a measure space (Ω , Σ , µ )with χ Ω ∈ X ( µ ) and let T : X ( µ ) → E be a Banach space valued operator. Itis known that if T is p -th power factorable then the largest function space towhich T can be extended preserving p -th power factorability is given by thespace L p ( m T ) of p -integrable functions with respect to m T , where m T : Σ → E is the vector measure associated to T via m T ( A ) = T ( χ A ). In this paper weextend this result by removing the restriction χ Ω ∈ X ( µ ). In this generalcase, by considering m T defined on a certain δ -ring, we show that the optimaldomain for T is the space L p ( m T ) ∩ L ( m T ). We apply the obtained resultsto the particular case when T is a map between sequence spaces defined by aninfinite matrix. Introduction
Although the concept of p -th power factorable operator have previously beenused as a tool in operator theory, it was introduced explicitly in [19, § , Σ , µ ) and a Banach function space X ( µ ) of ( µ -a.e. classes of) Σ-measurable functions such that χ Ω ∈ X ( µ ), for 1 ≤ p < ∞ , a Banach space valuedoperator T : X ( µ ) → E is p -th power factorable if there is a continuous extensionof T to the p -th power space X ( µ ) p of X ( µ ). This is equivalent to the existenceof a constant C > k T ( f ) k ≤ C k | f | p k pX ( µ ) = C k f k X ( µ ) p for all f ∈ X ( µ ). The main characterization of this class of operators establishesthat any of them can be extended to an space L p of a vector measure m T : Σ → E associated to T via m T ( A ) = T ( χ A ) and the extension is maximal. Note that the Date : August 11, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Integration with respect to a vector measure defined on a δ -ring,optimal extension of an operator, p -th power factorable operator, quasi-Banach function space.The first author gratefully acknowledge the support of the Ministerio de Econom´ıa y Compet-itividad (project condition χ Ω ∈ X ( µ ) is necessary for a correct definition of p -th power factorableoperator (i.e. X ( µ ) ⊂ X ( µ ) p ) and for m T to be well defined.Several applications are shown also in [19, § L q ( µ ) (Maurey-Rosenthal type theorems) and in harmonicanalysis (Fourier transform and convolution operators). After that, p -th powerfactorable operators have turned out to be a useful tool for the study of differentproblems in mathematical analysis, regarding for example Banach space interpo-lation theory [6], differential equations [10], description of maximal domains forseveral classes of operators [12], factorization of kernel operators [13] or adjointoperators [11].The requirement χ Ω ∈ X ( µ ) excludes basic spaces as L q (0 , ∞ ) or ℓ q . Althoughthese spaces can be represented as spaces satisfying the needed requirement (forinstance L q (0 , ∞ ) is isometrically isomorphic to L q ( e − x dx ) via the multiplicationoperator induced by e xq ), to use such a representation provides some kind of fac-torization for T but not genuine extensions.The aim of this paper is to extend the results on maximal extensions of p -th powerfactorable operators to quasi-Banach spaces X ( µ ) which do not necessary contain χ Ω . Also we will consider p to be any positive number removing the restriction p ≥
1. The first problem is the definition of p -th power factorable operator, as in generalthe containment X ( µ ) ⊂ X ( µ ) p does not hold. This can be solved by replacing X ( µ ) p by the sum X ( µ ) p + X ( µ ). The second problem is the definition of the vectormeasure m T associated to T . The technique to overcome this obstacle consists ofconsidering m T defined on the δ -ring Σ X ( µ ) = (cid:8) A ∈ Σ : χ A ∈ X ( µ ) (cid:9) instead of the σ -algebra Σ. We will see that actually no topology is needed on X ( µ ) to extend T : X ( µ ) → E , it suffices an ideal structure on X ( µ ) and a certain property on T which relates the µ -a.e. pointwise order of X ( µ ) and the weak topology of E . Thisproperty, called order-w continuity, is the minimal condition for m T to be a vectormeasure.The paper is organized as follows. Section 2 is devoted to establish the notationand to state the results on ideal function spaces, quasi-Banach function spaces andintegration with respect to a vector measure defined on a δ -ring, which will be usealong this work. For the aim of completeness, we include the proof of some relevantfacts. In Section 3 we show that every order-w continuous operator T defined onan ideal function space X ( µ ), can be extended to the space L ( m T ) of integrablefunctions with respect to m T and this space is the largest one to which T can beextended as an order-w continuous operator (Theorem 3.2). Section 4 deals withoperators T which are p -th power factorable with an order-w continuous extension,that is, there is an order-w continuous extension of T to the space X ( µ ) p + X ( µ ).We prove that the space L p ( m T ) ∩ L ( m T ) is the optimal domain for T preservingthe property of being p -th power factorable with an order-w continuous extension PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 3 (Theorem 4.2). In Sections 5 and 6 we endow X ( µ ) with a topology (namely, X ( µ )will be a σ -order continuous quasi-Banach function space) and consider T to becontinuous. Results on maximal extensions analogous to the ones of the previoussections are obtain for continuity instead of order-w continuity (Theorems 5.1 and6.2). Finally, as an application of our results, in the last section we study whenan infinite matrix of real numbers defines a continuous linear operator from ℓ p intoany given sequence space. 2. Preliminaries
Ideal function spaces.
Let (Ω , Σ) be a fixed measurable space. For a mea-sure µ : Σ → [0 , ∞ ], we denote by L ( µ ) the space of all ( µ -a.e. classes of) Σ-measurable real valued functions on Ω. Given two set functions µ, λ : Σ → [0 , ∞ ]we will write λ ≪ µ if µ ( A ) = 0 implies λ ( A ) = 0. We will say that µ and λ are equivalent if λ ≪ µ and µ ≪ λ . In the case when µ and λ are two measures with λ ≪ µ , the map [ i ] : L ( µ ) → L ( λ ) which takes a µ -a.e. class in L ( µ ) representedby f into the λ -a.e. class represented by the same f , is a well defined linear map.In order to simplify notation [ i ]( f ) will be denoted again as f . Note that if λ and µ are equivalent then L ( µ ) = L ( λ ) and [ i ] is the identity map i .An ideal function space (briefly, i.f.s.) is a vector space X ( µ ) ⊂ L ( µ ) satisfyingthat if f ∈ X ( µ ) and g ∈ L ( µ ) with | g | ≤ | f | µ -a.e. then g ∈ X ( µ ). We will saythat X ( µ ) has the σ -property if there exists (Ω n ) ⊂ Σ such that Ω = ∪ Ω n and χ Ω n ∈ X ( µ ) for all n . For instance, this happens if there is some g ∈ X ( µ ) with g > µ -a.e. Lemma 2.1.
Let X ( µ ) be an i.f.s. satisfying the σ -property. For every Σ –measurablefunction f : Ω → [0 , ∞ ) there exists ( f n ) ⊂ X ( µ ) such that ≤ f n ↑ f pointwise.Proof. Let (Ω n ) ⊂ Σ be the sequence given by the σ -property of X ( µ ) and let f : Ω → [0 , ∞ ) be a Σ–measurable function. Taking A n = ∪ nj =1 Ω j ∩ (cid:8) ω ∈ Ω : f ( ω ) ≤ n (cid:9) , we have that f n = f χ A n ∈ X ( µ ), as 0 ≤ f n ≤ nχ ∪ nj =1 Ω j pointwise, andthat f n ↑ f pointwise. (cid:3) The sum of two i.f.s.’ X ( µ ) and Y ( µ ) is the space defined as X ( µ ) + Y ( µ ) = (cid:8) f ∈ L ( µ ) : f = f + f µ -a.e. , f ∈ X ( µ ) , f ∈ Y ( µ ) (cid:9) . Proposition 2.2.
The sum X ( µ ) + Y ( µ ) of two i.f.s.’ is an i.f.s.Proof. Let f ∈ X ( µ ) + Y ( µ ) and g ∈ L ( µ ) be such that | g | ≤ | f | µ –a.e. Write f = f + f µ -a.e. with f ∈ X ( µ ) and f ∈ Y ( µ ) and denote A = (cid:8) ω ∈ Ω : | g ( ω ) | ≤| f ( ω ) | (cid:9) . Taking h = | g | χ A + | f | χ Ω \ A and h = ( | g | − | f | ) χ Ω \ A , we have that | g | = h + h with h ∈ X ( µ ) as 0 ≤ h ≤ | f | pointwise and h ∈ Y ( µ ) as 0 ≤ h ≤| f | µ -a.e. Now, denote B = (cid:8) ω ∈ Ω : g ( ω ) ≥ (cid:9) and take g = h (cid:0) χ B − χ Ω \ B ) and O. DELGADO AND E. A. S ´ANCHEZ P´EREZ g = h (cid:0) χ B − χ Ω \ B ). Then, g = g + g with g ∈ X ( µ ) as | g | = h and g ∈ Y ( µ )as | g | = h . So, g ∈ X ( µ ) + Y ( µ ). (cid:3) Let p ∈ (0 , ∞ ). The p -power of an i.f.s. X ( µ ) is the i.f.s. defined as X ( µ ) p = (cid:8) f ∈ L ( µ ) : | f | p ∈ X ( µ ) (cid:9) . Lemma 2.3.
Let X ( µ ) be an i.f.s. For s, t ∈ (0 , ∞ ) and r = s + t , it follows thatif f ∈ X ( µ ) s and g ∈ X ( µ ) t then f g ∈ X ( µ ) r . In particular, if χ Ω ∈ X ( µ ) then X ( µ ) q ⊂ X ( µ ) p for all < p < q < ∞ .Proof. For the first part only note that for every a, b > a r b r ≤ rs a s + rt b t . (2.1)For the second part take r = p , s = q and t = pqq − p . Then, if f ∈ X ( µ ) q , since χ Ω ∈ X ( µ ) t , we have that f = f χ Ω ∈ X ( µ ) p . (cid:3) Recall that a quasi-norm on a real vector space X is a non-negative real map k · k X on X satisfying(i) k x k X = 0 if and only if x = 0,(ii) k αx k X = | α | · k x k X for all α ∈ R and x ∈ X , and(iii) there exists a constant K ≥ k x + y k X ≤ K ( k x k X + k y k X ) for all x, y ∈ X .A quasi-norm k · k X induces a metric topology on X in which a sequence ( x n )converges to x if and only if k x − x n k X →
0. If X is complete under this topologythen it is called a quasi-Banach space ( Banach space if K = 1). A linear map T : X → Y between quasi-Banach spaces is continuous if and only if there exists aconstant M > k T ( x ) k Y ≤ M k x k X for all x ∈ X . For issues related toquasi-Banach spaces see [14].A quasi-Banach function space (quasi-B.f.s. for short) is a i.f.s. X ( µ ) which is alsoa quasi-Banach space with a quasi-norm k·k X ( µ ) compatible with the µ -a.e. pointwiseorder , that is, if f, g ∈ X ( µ ) are such that | f | ≤ | g | µ -a.e. then k f k X ( µ ) ≤ k g k X ( µ ) .When the quasi-norm is a norm, X ( µ ) is called a Banach function space (B.f.s.).Note that every quasi-B.f.s. is a quasi-Banach lattice for the µ -a.e. pointwise ordersatisfying that if f n → f in quasi-norm then there exists a subsequence f n j → fµ -a.e. Also note that every positive linear operator between quasi-Banach latticesis continuous, see the argument given in [16, p. 2] for Banach lattices which canbe adapted for quasi-Banach spaces. Then all “ inclusions” of the type [ i ] betweenquasi-B.f.s.’ are continuous.A quasi-B.f.s. X ( µ ) is said to be σ -order continuous if for every ( f n ) ⊂ X ( µ )with f n ↓ µ -a.e. it follows that k f n k X ↓ PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 5 It is routine to check that the intersection X ( µ ) ∩ Y ( µ ) of two quasi-B.f.s.’ (B.f.s.’) X ( µ ) and Y ( µ ) is a quasi-B.f.s. (B.f.s.) endowed with the quasi-norm (norm) k f k X ( µ ) ∩ Y ( µ ) = max (cid:8) k f k X ( µ ) , k f k Y ( µ ) (cid:9) . Moreover, if X ( µ ) and Y ( µ ) are σ -order continuous then X ( µ ) ∩ Y ( µ ) is σ -ordercontinuous. Proposition 2.4.
The sum X ( µ ) + Y ( µ ) of two quasi-B.f.s.’ (B.f.s.’) X ( µ ) and Y ( µ ) is a quasi-B.f.s. (B.f.s.) endowed with the quasi-norm (norm) k f k X ( µ )+ Y ( µ ) = inf (cid:0) k f k X ( µ ) + k f k Y ( µ ) (cid:1) , where the infimum is taken over all possible representations f = f + f µ -a.e. with f ∈ X ( µ ) and f ∈ Y ( µ ) . Moreover, if X ( µ ) and Y ( µ ) are σ -order continuousthen X ( µ ) + Y ( µ ) is also σ -order continuous.Proof. From Proposition 2.2 we have that X ( µ )+ Y ( µ ) is a i.f.s. Even more, lookingat the proof we see that for every f ∈ X ( µ ) + Y ( µ ) and g ∈ L ( µ ) with | g | ≤ | f | µ -a.e., if f = f + f µ -a.e. with f ∈ X ( µ ) and f ∈ Y ( µ ) then there exist g ∈ X ( µ )and g ∈ Y ( µ ) such that | g i | ≤ | f i | µ -a.e. and g = g + g . Then, k g k X ( µ )+ Y ( µ ) ≤ k g k X ( µ ) + k g k Y ( µ ) ≤ k f k X ( µ ) + k f k Y ( µ ) and so, taking infimum over all possible representations f = f + f µ -a.e. with f ∈ X ( µ ) and f ∈ Y ( µ ), it follows that k g k X ( µ )+ Y ( µ ) ≤ k f k X ( µ )+ Y ( µ ) . Hence, k · k X ( µ )+ Y ( µ ) is compatible with the µ -a.e. pointwise order.The proof of the fact that k ·k X ( µ )+ Y ( µ ) is a quasi-norm for which X ( µ )+ Y ( µ ) iscomplete is similar to the one given in [1, §
3, Theorem 1.3] for compatible couplesof Banach spaces.Suppose that X ( µ ) and Y ( µ ) are σ -order continuous. Let ( f n ) ⊂ X ( µ ) + Y ( µ )be such that f n ↓ µ -a.e. Consider f = g + h µ -a.e. with g ∈ X ( µ ) and h ∈ Y ( µ ).We can rewrite f = f + f with f ∈ X ( µ ), f ∈ Y ( µ ) and f , f ≥ µ -a.e. Thiscan be done by taking A = (cid:8) ω ∈ Ω : f ( ω ) ≤ | g ( ω ) | (cid:9) , f = f χ A + | g | χ Ω \ A and f = ( f − | g | ) χ Ω \ A . Note that f ∈ X ( µ ) as 0 ≤ f ≤ | g | µ -a.e. and f ∈ Y ( µ )as 0 ≤ f ≤ | h | µ -a.e. Since 0 ≤ f ≤ f µ -a.e., looking again at the proofof Proposition 2.2 we see that there exist f ∈ X ( µ ) and f ∈ Y ( µ ) such that0 ≤ f i ≤ f i µ -a.e. and f = f + f µ -a.e. By induction we construct two µ -a.e.pointwise decreasing sequences of positive functions ( f n ) ⊂ X ( µ ) and ( f n ) ⊂ Y ( µ )such that f n = f n + f n . Note that f in ↓ µ -a.e. as 0 ≤ f in ≤ f n µ -a.e. Then, since X ( µ ) and Y ( µ ) are σ -order continuous, we have that k f n k X ( µ )+ Y ( µ ) ≤ k f n k X ( µ ) + k f n k Y ( µ ) → . (cid:3) O. DELGADO AND E. A. S ´ANCHEZ P´EREZ
Let p ∈ (0 , ∞ ). The p -power X ( µ ) p of a quasi-B.f.s. X ( µ ) is a quasi-B.f.s.endowed with the quasi-norm k f k X ( µ ) p = k | f | p k p X ( µ ) . Moreover, X ( µ ) p is σ -order continuous whenever X ( µ ) is so. Note that in the casewhen X ( µ ) is a B.f.s. and p ≥ k · k X ( µ ) p is a norm and so X ( µ ) p isa B.f.s. An exhaustive study of the space X ( µ ) p can be found in [19, § µ is finite and χ Ω ∈ X ( µ ). This study can be extended to our generalcase adapting the arguments with the natural modifications (note that our p -powershere are the p -th powers there).2.2. Integration with respect to a vector measure defined on a δ -ring. Let R be a δ -ring of subsets of a set Ω, that is, a ring closed under countableintersections. Measurability will be considered with respect to the σ -algebra R loc of all subsets A of Ω such that A ∩ B ∈ R for all B ∈ R . Let us write S ( R ) for thespace of all R -simple functions, that is, simple functions with support in R .A set function m : R → E with values in a Banach space E is said to be a vectormeasure if P m ( A n ) converges to m ( ∪ A n ) in E for every sequence of pairwisedisjoint sets ( A n ) ⊂ R with ∪ A n ∈ R .Consider first a real measure λ : R → R . The variation of λ is the measure | λ | : R loc → [0 , ∞ ] defined as | λ | ( A ) = sup n X | λ ( A j ) | : ( A j ) finite disjoint sequence in R ∩ A o . Note that | λ | is finite on R . The space L ( λ ) of integrable functions with respectto λ is defined as the classical space L ( | λ | ). The integral with respect to λ of ϕ = P nj =1 α j χ A j ∈ S ( R ) over A ∈ R loc is defined in the natural way by R A ϕ dλ = P nj =1 α j λ ( A j ∩ A ). The space S ( R ) is dense in L ( λ ), allowing to define the integralof f ∈ L ( λ ) over A ∈ R loc as R A f dλ = lim R A ϕ n dλ for any sequence ( ϕ n ) ⊂ S ( R )converging to f in L ( λ ).Let now m : R → E be a vector measure. The semivariation of m is the setfunction k m k : R loc → [0 , ∞ ] defined by k m k ( A ) = sup x ∗ ∈ B E ∗ | x ∗ m | ( A ) . Here, B E ∗ is the closed unit ball of the dual space E ∗ of E and | x ∗ m | is the variationof the real measure x ∗ m given by the composition of m with x ∗ . A set A ∈ R loc is m -null if k m k ( A ) = 0, or equivalently, if m ( B ) = 0 for all B ∈ R ∩ A . From [2,Theorem 3.2], there always exists a measure η : R loc → [0 , ∞ ] equivalent to k m k ,that is m and η have the same null sets. Let us denote L ( m ) = L ( η ).The space L ( m ) of integrable functions with respect to m is defined as the spaceof functions f ∈ L ( m ) satisfying that(i) f ∈ L ( x ∗ m ) for every x ∗ ∈ E ∗ , and PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 7 (ii) for each A ∈ R loc there exists x A ∈ E such that x ∗ ( x A ) = Z A f dx ∗ m, for every x ∗ ∈ E ∗ . The vector x A is unique and will be denoted by R A f dm . The space L ( m ) is a σ -order continuous B.f.s. related to the measure space (Ω , R loc , η ), with norm k f k L ( m ) = sup x ∗ ∈ B E ∗ Z Ω | f | d | x ∗ m | . Moreover S ( R ) is dense in L ( m ). Note that R A ϕ dm = P nj =1 α j m ( A j ∩ A ) forevery ϕ = P nj =1 α j χ A j ∈ S ( R ) and A ∈ R loc .The integration operator I m : L ( m ) → E defined by I m ( f ) = R Ω f dm is acontinuous linear operator with k I m ( f ) k E ≤ k f k L ( m ) . Even more,12 k f k L ( m ) ≤ sup A ∈R k I m ( f χ A ) k E ≤ k f k L ( m ) (2.2)for all f ∈ L ( m ).Let p ∈ (0 , ∞ ). We denote by L p ( m ) the p -power of L ( m ), that is, L p ( m ) = (cid:8) f ∈ L ( m ) : | f | p ∈ L ( m ) (cid:9) . Then L p ( m ) is a quasi-B.f.s. with the quasi-norm k f k L p ( m ) = k | f | p k /pL ( m ) . In thecase when p ≥
1, we have that k · k L p ( m ) is a norm and so L p ( m ) is a B.f.s.These and other issues concerning integration with respect to a vector measuredefined on a δ -ring can be found in [15], [17], [18], [7], [5] and [3].3. Optimal domain for order-w continuous operators on a i.f.s.
Let X ( µ ) be a i.f.s. satisfying the σ -property (recall: Ω = ∪ Ω n with χ Ω n ∈ X ( µ )for all n ) and consider the δ –ringΣ X ( µ ) = (cid:8) A ∈ Σ : χ A ∈ X ( µ ) (cid:9) . The σ -property guarantees that Σ locX ( µ ) = Σ. Given a Banach space valued linearoperator T : X ( µ ) → E , we define the finitely additive set function m T : Σ X ( µ ) → E by m T ( A ) = T ( χ A ).We will say that T is order-w continuous if T ( f n ) → T ( f ) weakly in E whenever f n , f ∈ X ( µ ) are such that 0 ≤ f n ↑ f µ –a.e. Proposition 3.1. If T is order-w continuous, then m T is a vector measure satis-fying that [ i ] : X ( µ ) → L ( m T ) is well defined and T = I m T ◦ [ i ] .Proof. Let ( A n ) ⊂ Σ X ( µ ) be a pairwise disjoint sequence with ∪ A n ∈ Σ X ( µ ) . Since T is order-w continuous, for any subsequence ( A n j ) we have that N X j =1 m T ( A n j ) = T ( χ ∪ Nj =1 A nj ) → T ( χ ∪ A nj ) = m T ( ∪ A n j ) O. DELGADO AND E. A. S ´ANCHEZ P´EREZ weakly in E . From the Orlicz-Pettis theorem (see [9, Corollary I.4.4]) it followsthat P m T ( A n ) is unconditionally convergent in norm to m T ( ∪ A n ). Thus, m T isa vector measure.Note that k m T k ≪ µ and so [ i ] : L ( µ ) → L ( m T ) is well defined. Also, notethat for every ϕ ∈ S (Σ X ( µ ) ) we have that I m T ( ϕ ) = T ( ϕ ).Let f ∈ X ( µ ) be such that f ≥ µ -a.e. and take a sequence of Σ-simple functions0 ≤ ϕ n ↑ f µ -a.e. For each n we can write ϕ n = P mj =1 α j χ A j with ( A j ) mj =1 ⊂ Σbeing a pairwise disjoint sequence and α j > j . Since χ A j ≤ α − j ϕ n ≤ α − j fµ -a.e., we have that χ A j ∈ X ( µ ) and so ϕ n ∈ S (Σ X ( µ ) ). Fix x ∗ ∈ E ∗ . For every A ∈ Σ it follows that x ∗ T ( ϕ n χ A ) → x ∗ T ( f χ A ) as T is order-w continuous. Notethat x ∗ T ( ϕ n χ A ) = R A ϕ n dx ∗ m T and that 0 ≤ ϕ n ↑ f x ∗ m T -a.e. as | x ∗ m T | ≪k m T k ≪ µ . From [7, Proposition 2.3], we have that f ∈ L ( x ∗ m T ) and Z A f dx ∗ m T = lim n →∞ Z A ϕ n dx ∗ m T = lim n →∞ x ∗ T ( ϕ n χ A ) = x ∗ T ( f χ A ) . Therefore, f ∈ L ( m T ) and I m T ( f ) = T ( f ).For a general f ∈ X ( µ ), the result follows by taking the positive and negativeparts of f . (cid:3) For the case when X ( µ ) is a B.f.s., Proposition 3.1 and the next Theorem 3.2can be deduced from [8, Proposition 2.3] and [4, Proposition 4]. The proofs givenhere are more direct and are valid for general i.f.s.’. Theorem 3.2.
Suppose that T is order-w continuous. Then, T factors as X ( µ ) T / / [ i ] $ $ EL ( m T ) I mT < < (3.1) with I m T being order-w continuous. Moreover, the factorization is optimal in thesense:If Z ( ξ ) is a i.f.s. such that ξ ≪ µ and X ( µ ) T / / [ i ] EZ ( ξ ) S > > (3.2) with S being an order-w continuous linearoperator = ⇒ [ i ] : Z ( ξ ) → L ( m T ) is welldefined and S = I m T ◦ [ i ] .Proof. The factorization (3.1) follows from Proposition 3.1. Note that the inte-gration operator I m T : L ( m T ) → E is order-w continuous, as it is continuous and L ( m T ) is σ -order continuous. PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 9 Let Z ( ξ ) satisfy (3.2). In particular, Z ( ξ ) satisfies the σ -property, as if χ A ∈ X ( µ ) then χ A ∈ Z ( ξ ). From Proposition 3.1 applied to the operator S : Z ( ξ ) → E ,we have that [ i ] : Z ( ξ ) → L ( m S ) is well defined and S = I m S ◦ [ i ]. Note thatΣ X ( µ ) ⊂ Σ Z ( ξ ) and m S ( A ) = S ( χ A ) = T ( χ A ) = m T ( A ) for all A ∈ Σ X ( µ ) , thatis, m T is the restriction of m S : Σ Z ( ξ ) → E to Σ X ( µ ) . Then, from [4, Lemma 3], itfollows that L ( m S ) = L ( m T ) and I m S = I m T . (cid:3) We can rewrite Theorem 3.2 in terms of optimal domain.
Corollary 3.3.
Suppose that T is order-w continuous. Then L ( m T ) is the largesti.f.s. to which T can be extended as an order-w continuous operator still with valuesin E . Moreover, the extension of T to L ( m T ) is given by the integration operator I m T . Optimal domain for p -th power factorable operators on a i.f.s.with an order-w continuous extension Let X ( µ ) be a i.f.s. satisfying the σ -property and let T : X ( µ ) → E be a linearoperator with values in a Banach space E .For p ∈ (0 , ∞ ), we call T p -th power factorable with an order-w continuousextension if there is an order-w continuous linear extension of T to X ( µ ) p + X ( µ ),i.e. T factors as X ( µ ) T / / i & & EX ( µ ) p + X ( µ ) S with S being an order-w continuous linear operator.Note that in the case when χ Ω ∈ X ( µ ), from Lemma 2.3, if 1 < p we havethat X ( µ ) ⊂ X ( µ ) p and so X ( µ ) p + X ( µ ) = X ( µ ) p . Similarly, if p ≤ X ( µ ) p + X ( µ ) = X ( µ ), but hence to say that T is p -th power factorable with anorder-w continuous extension is just to say that T is order-w continuous. Proposition 4.1.
The following statements are equivalent: (a) T is p -th power factorable with an order-w continuous extension. (b) T is order-w continuous and [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is well defined. (c) T is order-w continuous and [ i ] : X ( µ ) → L p ( m T ) ∩ L ( m T ) is well defined.Moreover, if (a)-(c) holds, the extension of T to X ( µ ) p + X ( µ ) coincides withintegration operator I m T ◦ [ i ] .Proof. (a) ⇒ (b) Note that T is order-w continuous as it has an order-w continuousextension. Let S : X ( µ ) p + X ( µ ) → E be an order-w continuous linear operatorextending T . Then, from Theorem 3.2, it follows that [ i ] : X ( µ ) p + X ( µ ) → L ( m T )is well defined and S = I m T ◦ [ i ]. (b) ⇔ (c) Since T is is order-w continuous, by Proposition 3.1 we always have that[ i ] : X ( µ ) → L ( m T ) is well defined. Suppose that [ i ] : X ( µ ) p + X ( µ ) → L ( m T )is well defined. If f ∈ X ( µ ), since | f | p ∈ X ( µ ) p ⊂ X ( µ ) p + X ( µ ), we have that | f | p ∈ L ( m T ) and so f ∈ L p ( m T ). Then f ∈ L p ( m T ) ∩ L ( m T ). Conversely,suppose that [ i ] : X ( µ ) → L p ( m T ) ∩ L ( m T ) is well defined. Let f ∈ X ( µ ) p + X ( µ )and write f = f + f µ -a.e. with f ∈ X ( µ ) p and f ∈ X ( µ ). Since | f | p ∈ X ( µ )we have that | f | p ∈ L p ( m T ) ∩ L ( m T ) ⊂ L p ( m T ) and so f ∈ L ( m T ). Then, f ∈ L ( m T ) as f ∈ L ( m T ).(b) ⇒ (a) From Proposition 3.1 and since [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is welldefined, we have that the operator I m T ◦ [ i ] extends T to X ( µ ) p + X ( µ ). Moreover,the extension I m T ◦ [ i ] : X ( µ ) p + X ( µ ) → E is order-w continuous as the integrationoperator I m T : L ( m T ) → E is so. (cid:3) In the case when χ Ω ∈ X ( µ ) and T is order-w continuous, from Proposition 3.1,we have that χ Ω ∈ L ( m T ). So, from Lemma 2.3, if p > L p ( m T ) ⊂ L ( m T )and hence L p ( m T ) ∩ L ( m T ) = L p ( m T ). If p ≤ L p ( m T ) ∩ L ( m T ) = L ( m T ),but hence, as commented before, T being p -th power factorable with an order-wcontinuous extension is just T being order-w continuous. Theorem 4.2.
Suppose that T is p -th power factorable with an order-w continuousextension. Then, T factors as X ( µ ) T / / [ i ] ' ' EL p ( m T ) ∩ L ( m T ) I mT (4.1) with I m T being p -th power factorable with an order-w continuous extension. More-over, the factorization is optimal in the sense:If Z ( ξ ) is a i.f.s. such that ξ ≪ µ and X ( µ ) T / / [ i ] EZ ( ξ ) S > > (4.2) with S being a p -th power factorable lin-ear operator with an order-w continuousextension = ⇒ [ i ] : Z ( ξ ) → L p ( m T ) ∩ L ( m T ) is well defined and S = I m T ◦ [ i ] .Proof. The factorization (4.1) follows from Propositions 3.1 and 4.1. Note that L p ( m T ) ∩ L ( m T ) satisfies the σ -property as X ( µ ) does. Let us see that theoperator I m T : L p ( m T ) ∩ L ( m T ) → E is p -th power factorable with an order-wcontinuous extension by ussing Proposition 4.1.(c). This operator is order-w con-tinuous as the integration operator I m T : L ( m T ) → E is so. On other hand, since PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 11 Σ X ( µ ) ⊂ Σ L p ( m T ) ∩ L ( m T ) and m I mT ( A ) = I m T ( χ A ) = T ( χ A ) = m T ( A ) for all A ∈ Σ X ( µ ) (i.e. m T is the restriction of m I mT : Σ L p ( m T ) ∩ L ( m T ) → E to Σ X ( µ ) ),from [4, Lemma 3], it follows that L ( m I mT ) = L ( m T ). Then,[ i ] : L p ( m T ) ∩ L ( m T ) → L p ( m I mT ) ∩ L ( m I mT ) = L p ( m T ) ∩ L ( m T )is well defined.Let Z ( ξ ) satisfy (4.2). In particular, Z ( ξ ) has the σ -property. Applying Propo-sition 4.1 to the operator S : Z ( ξ ) → E , we have that [ i ] : Z ( ξ ) → L p ( m S ) ∩ L ( m S )is well defined and S = I m S ◦ [ i ]. Since Σ X ( µ ) ⊂ Σ Z ( ξ ) and m S ( A ) = m T ( A )for all A ∈ Σ X ( µ ) , from [4, Lemma 3], it follows that L ( m S ) = L ( m T ) and I m S = I m T . (cid:3) Rewriting Theorem 4.2 in terms of optimal domain we obtain the followingconclusion.
Corollary 4.3.
Suppose that T is p -th power factorable with an order-w continuousextension. Then L p ( m T ) ∩ L ( m T ) is the largest i.f.s. to which T can be extendedas a p -th power factorable operator with an order-w continuous extension, still withvalues in E . Moreover, the extension of T to L p ( m T ) ∩ L ( m T ) is given by theintegration operator I m T . Optimal domain for continuous operators on a quasi-B.f.s.
Let X ( µ ) be a quasi-B.f.s. satisfying the σ -property and let T : X ( µ ) → E be alinear operator with values in a Banach space E . Theorem 5.1.
Suppose that X ( µ ) is σ -order continuous and T is continuous.Then, T factors as X ( µ ) T / / [ i ] $ $ EL ( m T ) I mT < < (5.1) with I m T being continuous. Moreover, the factorization is optimal in the sense:If Z ( ξ ) is a σ -order continuous quasi-B.f.s.such that ξ ≪ µ and X ( µ ) T / / [ i ] EZ ( ξ ) S > > (5.2) with S being a continuous linear operator = ⇒ [ i ] : Z ( ξ ) → L ( m T ) is welldefined and S = I m T ◦ [ i ] . Proof.
Since X ( µ ) is σ -order continuous and T is continuous we have that T isorder-w continuous and so the factorization (5.1) follows from Theorem 3.2. Recallthat L ( m T ) is σ -order continuous and I m T is continuous.Let Z ( ξ ) satisfy (5.2). In particular S is order-w continuous. From Theorem 3.2we have that [ i ] : Z ( ξ ) → L ( m T ) is well defined and S = I m T ◦ [ i ]. (cid:3) Corollary 5.2.
Suppose that X ( µ ) is σ -order continuous and T is continuous.Then L ( m T ) is the largest σ -order continuous quasi-B.f.s. to which T can be ex-tended as a continuous operator still with values in E . Moreover, the extension of T to L ( m T ) is given by the integration operator I m T . Optimal domain for p -th power factorable operators on aquasi-B.f.s. with a continuous extension Let X ( µ ) be a quasi-B.f.s. satisfying the σ -property and let T : X ( µ ) → E be alinear operator with values in a Banach space E .For p ∈ (0 , ∞ ), we call T p -th power factorable with a continuous extension ifthere is a continuous linear extension of T to X ( µ ) p + X ( µ ), i.e. T factors as X ( µ ) T / / i & & EX ( µ ) p + X ( µ ) S with S being a continuous linear operator.Note that in the case when χ Ω ∈ X ( µ ) and 1 < p , from Lemma 2.3, it followsthat X ( µ ) p + X ( µ ) = X ( µ ) p . Then our definition of p -th power factorable operatorwith a continuous extension coincides with the one given in [19, Definition 5.1]. If p ≤
1, since X ( µ ) p + X ( µ ) = X ( µ ), to say that T is p -th power factorable with acontinuous extension is just to say that T is continuous. Proposition 6.1.
Suppose that X ( µ ) is σ -order continuous. Then, the followingstatements are equivalent: (a) T is p -th power factorable with a continuous extension. (b) T is p -th power factorable with an order-w continuous extension. (c) T is order-w continuous and [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is well defined. (d) T is order-w continuous and [ i ] : X ( µ ) → L p ( m T ) ∩ L ( m T ) is well defined. (e) There exists
C > such that k T ( f ) k E ≤ C k f k X ( µ ) p + X ( µ ) for all f ∈ X ( µ ) .Moreover, if (a)-(e) holds, the extension of T to X ( µ ) p + X ( µ ) coincides with theintegration operator I m T ◦ [ i ] .Proof. (a) ⇒ (b) Let S : X ( µ ) p + X ( µ ) → E be a continuous linear operator ex-tending T . From Proposition 2.4 we have that X ( µ ) p + X ( µ ) is σ -order continuous PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 13 and so S is order-w continuous. Then, T is p -th power factorable with an order-wcontinuous extension.(b) ⇔ (c) ⇔ (d) and the fact that the extension of T to X ( µ ) p + X ( µ ) coincideswith the integration operator I m T ◦ [ i ] follow from Proposition 4.1.(c) ⇒ (e) The operator [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is continuous as it ispositive. Then, there exists a constant C > k f k L ( m T ) ≤ C k f k X ( µ ) p + X ( µ ) for all f ∈ X ( µ ) p + X ( µ ). Since I m T extends T to L ( m T ), it follows that k T ( f ) k E = k I m T ( f ) k E ≤ k f k L ( m T ) ≤ C k f k X ( µ ) p + X ( µ ) for all f ∈ X ( µ ).(e) ⇒ (a) Let 0 ≤ f ∈ X ( µ ) p + X ( µ ). From Lemma 2.1, there exists ( f n ) ⊂ X ( µ )such that 0 ≤ f n ↑ f µ -a.e. Since X ( µ ) p + X ( µ ) is σ -order continuous, it followsthat f n → f in the quasi-norm of X ( µ ) p + X ( µ ). Then, since k T ( f n ) − T ( f m ) k E = k T ( f n − f m ) k E ≤ C k f n − f m k X ( µ ) p + X ( µ ) , we have that (cid:0) T ( f n ) (cid:1) converges to some element e ∈ E . Define S ( f ) = e . Notethat if ( g n ) ⊂ X ( µ ) is another sequence such that 0 ≤ g n ↑ f µ -a.e., then k T ( f n ) − T ( g n ) k E ≤ C k f n − g n k X ( µ ) p + X ( µ ) ≤ CK (cid:16) k f n − f k X ( µ ) p + X ( µ ) + k f − g n k X ( µ ) p + X ( µ ) (cid:17) , where K is the constant satisfying the property (iii) of the quasi-norm k·k X ( µ ) p + X ( µ ) ,and so S is well defined. Also note that k S ( f ) k E ≤ k S ( f ) − T ( f n ) k E + k T ( f n ) k E ≤ k S ( f ) − T ( f n ) k E + C k f n k X ( µ ) p + X ( µ ) ≤ k S ( f ) − T ( f n ) k E + C k f k X ( µ ) p + X ( µ ) for all n ≥
1, and thus k S ( f ) k E ≤ C k f k X ( µ ) p + X ( µ ) .For a general f ∈ X ( µ ) p + X ( µ ), define S ( f ) = S ( f + ) − S ( f − ) where f + and f − are the positive and negative parts of f respectively. It follows that S is linearand S ( f ) = T ( f ) for all f ∈ X ( µ ). Moreover, for every f ∈ X ( µ ) p + X ( µ ) we havethat k S ( f ) k E ≤ k S ( f + ) k E + k S ( f − ) k E ≤ C k f + k X ( µ ) p + X ( µ ) + C k f − k X ( µ ) p + X ( µ ) ≤ C k f k X ( µ ) p + X ( µ ) . an so S is continuous. Hence, T is p -th power factorable with a continuous exten-sion. (cid:3) In the case when µ is finite, χ Ω ∈ X ( µ ) and p ≥
1, the equivalences (a) ⇔ (c) ⇔ (d) ⇔ (e) of Proposition 6.1 are proved in [19, Theorem 5.7]. Here we haveincluded a more detailed proof for the general case. Theorem 6.2.
Suppose that X ( µ ) is σ -order continuous and T is p -th power fac-torable with a continuous extension. Then, T factors as X ( µ ) T / / [ i ] ' ' EL p ( m T ) ∩ L ( m T ) I mT (6.1) with I m T being p -th power factorable with a continuous extension. Moreover, thefactorization is optimal in the sense:If Z ( ξ ) is a σ -order continuous quasi-B.f.s. such that ξ ≪ µ and X ( µ ) T / / [ i ] EZ ( ξ ) S > > (6.2) with S being a p -th power factorable lin-ear operator with a continuous extension = ⇒ [ i ] : Z ( ξ ) → L p ( m T ) ∩ L ( m T ) is well defined and S = I m T ◦ [ i ] .Proof. From Proposition 6.1 we have that T is p -th power factorable with an order-w continuous extension. Then, from Theorem 4.2, the factorization (6.1) holds and I m T : L p ( m T ) ∩ L ( m T ) → E is p -th power factorable with an order-w continu-ous extension. Noting that the space L p ( m T ) ∩ L ( m T ) is σ -order continuous (as L ( m T ) is so) and satisfies the σ -property (as X ( µ ) does), from Proposition 6.1 itfollows that I m T : L p ( m T ) ∩ L ( m T ) → E is p -th power factorable with a continuousextension.Let Z ( ξ ) satisfy (6.2), in particular it satisfies the σ -property. Again Proposition6.1 gives that S is p -th power factorable with an order-w continuous extension. So,from Theorem 4.2, it follows that [ i ] : Z ( ξ ) → L p ( m T ) ∩ L ( m T ) is well defined and S = I m T ◦ [ i ]. (cid:3) Corollary 6.3.
Suppose that X ( µ ) is σ -order continuous and T is p -th powerfactorable with a continuous extension. Then L p ( m T ) ∩ L ( m T ) is the largest σ -order continuous quasi-B.f.s. to which T can be extended as a p -th power factorableoperator with a continuous extension, still with values in E . Moreover, the extensionof T to L p ( m T ) ∩ L ( m T ) is given by the integration operator I m T . In the case when µ is finite, χ Ω ∈ X ( µ ) and p ≥
1, Corollary 6.3 is proved in[19, Theorem 5.11].
PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 15 Application: extension for operators defined on ℓ Consider the measure space ( N , P ( N ) , c ) where c is the counting measure on N .Note that a property holds c -a.e. if and only if it holds pointwise and that thespace L ( c ) coincides with the space ℓ of all real sequences. Consider the space ℓ = L ( c ), which is σ -order continuous and has the σ -property. The δ -ring P ( N ) ℓ is just the set P F ( N ) of all finite subsets of N .Let T : ℓ → E be a continuous linear operator with values in a Banach space E .Denote e n = χ { n } and assume that T ( e n ) = 0 for all n . This assumption seems tobe natural since if T ( e n ) = 0 then the n -th coordinate is not involved in the action of T . Hence, the vector measure m T : P F ( N ) → E associated to T by m T ( A ) = T ( χ A )is equivalent to c and so L ( m T ) ⊂ ℓ . We will write ℓ ( m T ) = L ( m T ). Remark . By Theorem 5.1 we have that T can be extended as ℓ T / / i " " Eℓ ( m T ) I mT < < and ℓ ( m T ) is the largest σ -order continuous quasi-B.f.s. to which T can be extendedas a continuous operator.Let p >
1. We have that T is p -th power factorable with a continuous extensionif there is an extension S as ℓ T / / i (cid:31) (cid:31) Eℓ p S ? ? with S being a continuous linear operator. Note that p ≤ ℓ p ⊂ ℓ and so the extension of T to the sum ℓ p + ℓ is just the sameoperator T . Applying Proposition 6.1 in the context of this section we obtain thefollowing result. Proposition 7.2.
The following statements are equivalent: (a) T is p -th power factorable with a continuous extension. (b) ℓ p ⊂ ℓ ( m T ) . (c) ℓ ⊂ ℓ p ( m T ) ∩ ℓ ( m T ) . (d) There exists
C > such that (cid:13)(cid:13)(cid:13) X j ∈ M x j T ( e j ) (cid:13)(cid:13)(cid:13) E ≤ C (cid:16) X j ∈ M x pj (cid:17) p for all M ∈ P F ( N ) and ( x j ) j ∈ M ⊂ [0 , ∞ ) . Proof.
From Proposition 6.1, we only have to prove that condition (d) is equivalentto the following condition:(d’) There exists
C > k T ( x ) k E ≤ C k x k ℓ p for all x ∈ ℓ .If (d’) holds, we obtain (d) by taking in (d’) the element x = P j ∈ M x j e j ∈ ℓ for every M ∈ P F ( N ) and ( x j ) j ∈ M ⊂ [0 , ∞ ).Suppose that (d) holds. Let 0 ≤ x = ( x n ) ∈ ℓ and take y k = P kj =1 x j e j . Since y k ↑ x pointwise, ℓ is σ -order continuous and T is continuous, we have that k T ( x ) k E = lim k T ( y k ) k E = lim (cid:13)(cid:13)(cid:13) k X j =1 x j T ( e j ) (cid:13)(cid:13)(cid:13) E ≤ C lim (cid:16) k X j =1 x pj (cid:17) p = C k x k ℓ p . For a general x ∈ ℓ , (d’) follows by taking the positive and negative parts of x . (cid:3) Remark . Note that if T is p -th power factorable with a continuous extensionthen the integration operator I m T extends T to ℓ p and, from Theorem 6.2, T factorsoptimally as ℓ T / / i & & Eℓ p ( m T ) ∩ ℓ ( m T ) I mT with I m T being p -th power factorable with a continuous extension.Now a natural question arises: When ℓ p ( m T ) ∩ ℓ ( m T ) is equal to ℓ p ( m T ) or ℓ ( m T )? For asking this question we introduce the following class of operators.Let 0 < r < ∞ . We say that T is r -power dominated if there exists C > (cid:13)(cid:13)(cid:13) X j ∈ M x rj T ( e j ) (cid:13)(cid:13)(cid:13) r E ≤ C sup N ⊂ M (cid:13)(cid:13)(cid:13) X j ∈ N x j T ( e j ) (cid:13)(cid:13)(cid:13) E for every M ∈ P F ( N ) and ( x j ) j ∈ M ∈ [0 , ∞ ). Note that in the case when E is aBanach lattice and T is positive we have thatsup N ⊂ M (cid:13)(cid:13)(cid:13) X j ∈ N x j T ( e j ) (cid:13)(cid:13)(cid:13) E = (cid:13)(cid:13)(cid:13) X j ∈ M x j T ( e j ) (cid:13)(cid:13)(cid:13) E . Lemma 7.4.
The containment ℓ ( m T ) ⊂ ℓ r ( m T ) holds if and only if T is r -powerdominated.Proof. Suppose that ℓ ( m T ) ⊂ ℓ r ( m T ). Since the containment is continuous (as it ispositive), there exists C > k x k ℓ r ( m T ) ≤ C k x k ℓ ( m T ) for all x ∈ ℓ ( m T ).For every M ∈ P F ( N ) and ( x j ) j ∈ M ∈ [0 , ∞ ), we consider x = P j ∈ M x j e j ∈ ℓ .Noting that x r = P j ∈ M x rj e j ∈ ℓ , it follows that (cid:13)(cid:13)(cid:13) X j ∈ M x rj T ( e j ) (cid:13)(cid:13)(cid:13) r E = k T ( x r ) k r E = k I m T ( x r ) k r E ≤ k x r k r ℓ ( m T ) = k x k ℓ r ( m T ) ≤ C k x k ℓ ( m T ) ≤ C sup A ∈P F ( N ) k I m T ( xχ A ) k E , PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 17 where in the last inequality we have used (2.2). For every A ∈ P F ( N ) we have that xχ A = P j ∈ A ∩ M x j e j ∈ ℓ and so I m T ( xχ A ) = T ( xχ A ) = P j ∈ A ∩ M x j T ( e j ). Then, (cid:13)(cid:13)(cid:13) X j ∈ M x rj T ( e j ) (cid:13)(cid:13)(cid:13) r E ≤ C sup A ∈P F ( N ) (cid:13)(cid:13)(cid:13) X j ∈ A ∩ M x j T ( e j ) (cid:13)(cid:13)(cid:13) E = 2 C sup N ⊂ M (cid:13)(cid:13)(cid:13) X j ∈ N x j T ( e j ) (cid:13)(cid:13)(cid:13) E . Conversely, suppose that T is r -power dominated and let x = ( x n ) ∈ ℓ ( m T ).Taking y k = P kj =1 | x j | r e j ∈ ℓ , for every k > ˜ k and A ∈ P F ( N ), we have that( y k − y ˜ k ) χ A = P j ∈ A ∩{ ˜ k +1 ,...,k } | x j | r e j and so (cid:13)(cid:13) T (cid:0) ( y k − y ˜ k ) χ A (cid:1)(cid:13)(cid:13) E = (cid:13)(cid:13)(cid:13) X j ∈ A ∩{ ˜ k +1 ,...,k } | x j | r T ( e j ) (cid:13)(cid:13)(cid:13) E ≤ C r sup N ⊂ A ∩{ ˜ k +1 ,...,k } (cid:13)(cid:13)(cid:13) X j ∈ N | x j | T ( e j ) (cid:13)(cid:13)(cid:13) rE = C r sup N ⊂ A ∩{ ˜ k +1 ,...,k } (cid:13)(cid:13)(cid:13) I m T (cid:16) X j ∈ N | x j | e j (cid:17)(cid:13)(cid:13)(cid:13) rE ≤ C r sup N ⊂ A ∩{ ˜ k +1 ,...,k } (cid:13)(cid:13)(cid:13) X j ∈ N | x j | e j (cid:13)(cid:13)(cid:13) rℓ ( m T ) ≤ C r (cid:13)(cid:13) ( y k ) r − ( y ˜ k ) r (cid:13)(cid:13) rℓ ( m T ) . For the last inequality note that ( y k ) r = P kj =1 | x j | e j and so X j ∈ N | x j | e j ≤ k X j =˜ k +1 | x j | e j = ( y k ) r − ( y ˜ k ) r for every N ⊂ A ∩ { ˜ k + 1 , ..., k } . Then, by using (2.2), we have that (cid:13)(cid:13) y k − y ˜ k (cid:13)(cid:13) ℓ ( m T ) ≤ A ∈P F ( N ) (cid:13)(cid:13) I m T (cid:0) ( y k − y ˜ k ) χ A (cid:1)(cid:13)(cid:13) E = 2 sup A ∈P F ( N ) (cid:13)(cid:13) T (cid:0) ( y k − y ˜ k ) χ A (cid:1)(cid:13)(cid:13) E ≤ C r (cid:13)(cid:13) ( y k ) r − ( y ˜ k ) r (cid:13)(cid:13) rℓ ( m T ) → k, ˜ k → ∞ since ( y k ) r ↑ | x | pointwise and ℓ ( m T ) is σ -order continuous. Hence, y k → z in ℓ ( m T ) for some z ∈ ℓ ( m T ). In particular, y k → z pointwise and so | x | r = z ∈ ℓ ( m T ) as y k ↑ | x | r pointwise. Therefore x ∈ ℓ r ( m T ). (cid:3) Lemma 7.5.
Let p > . If T is p -power dominated then it is p -th power factorablewith a continuous extension. Proof.
Let us use Proposition 7.2.(d). Given M ∈ P F ( N ) and ( x j ) j ∈ M ⊂ [0 , ∞ ),denoting by K the continuity constant of T , we have that (cid:13)(cid:13)(cid:13) X j ∈ M x j T ( e j ) (cid:13)(cid:13)(cid:13) E = (cid:13)(cid:13)(cid:13) X j ∈ M ( x pj ) p T ( e j ) (cid:13)(cid:13)(cid:13) E ≤ C p sup N ⊂ M (cid:13)(cid:13)(cid:13) X j ∈ N x pj T ( e j ) (cid:13)(cid:13)(cid:13) p E = C p sup N ⊂ M (cid:13)(cid:13)(cid:13) T (cid:16) X j ∈ N x pj e j (cid:17)(cid:13)(cid:13)(cid:13) p E ≤ C p K p sup N ⊂ M (cid:13)(cid:13)(cid:13) X j ∈ N x pj e j (cid:13)(cid:13)(cid:13) p ℓ = C p K p sup N ⊂ M (cid:16) X j ∈ N x pj (cid:17) p ≤ C p K p (cid:16) X j ∈ M x pj (cid:17) p . (cid:3) As a consequence of Remark 7.3, Lemma 7.4 and Lemma 7.5, we obtain thefollowing conclusion.
Corollary 7.6.
For p > we have that: (a) If T is p -power dominated and p -th power factorable with a continuous exten-sion, then T factors optimally as ℓ T / / i " " Eℓ p ( m T ) I mT < < with I m T being p -th power factorable with a continuous extension. (b) If T is p -power dominated, then T factors optimally as ℓ T / / i " " Eℓ ( m T ) I mT < < with I m T being p -th power factorable with a continuous extension. Consider now the case when E = ℓ ( c ) is a B.f.s. related to c such that ℓ ⊂ ℓ ( c ) ⊂ ℓ . Then ℓ ( c ) is a K¨othe function space in the sense of Lindenstrauss andTzafriri, see [16, p. 28-30]. For instance, ℓ ( c ) could be an ℓ q space with 1 ≤ q ≤ ∞ ,or a Lorentz sequence space ℓ q,r with 1 ≤ r ≤ q ≤ ∞ or an Orlicz sequence space ℓ ϕ with ϕ being an Orlicz function.Let us recall some facts about the K¨othe dual of an space ℓ ( c ). Denote the scalarproduct of two sequences x = ( x n ) , y = ( y n ) ∈ ℓ by (cid:0) x, y (cid:1) = X x n y n PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 19 provided the sum exists. The K¨othe dual of ℓ ( c ) is given by ℓ ( c ) ′ = n y ∈ ℓ : (cid:0) | x | , | y | (cid:1) < ∞ for all x ∈ ℓ ( c ) o . Note that χ A ∈ ℓ ( c ) ′ for all A ∈ P F ( N ). The space ℓ ( c ) ′ endowed with the norm k y k ℓ ( c ) ′ = sup x ∈ B ℓ ( c ) (cid:0) | x | , | y | (cid:1) is a B.f.s. in the sense of Lindenstrauss and Tzafriri. The map j : ℓ ( c ) ′ → ℓ ( c ) ∗ defined by h j ( y ) , x i = (cid:0) x, y (cid:1) for all y ∈ ℓ ( c ) ′ and x ∈ ℓ ( c ), is a linear isometry. Inparticular, convergence in norm of ℓ ( c ) implies pointwise convergence, as e n ∈ ℓ ( c ) ′ for all n . Note that ℓ ( c ) ⊂ ℓ ( c ) ′′ . The equality ℓ ( c ) = ℓ ( c ) ′′ holds with equalnorms if and only if ℓ ( c ) has the Fatou property , that is, if ( x k ) ⊂ ℓ ( c ) is such that0 ≤ x k ↑ x pointwise and sup k x k k ℓ ( c ) < ∞ then x ∈ ℓ ( c ) and k x k k ℓ ( c ) ↑ k x k ℓ ( c ) .Let M = ( a ij ) be an infinite matrix of real numbers and denote by C j the j -thcolumn of M . Assume C j = 0 for all j . Note that M x = (cid:16) X j a ij x j (cid:17) i for any x ∈ ℓ for which it is meaningful to do so. Proposition 7.7.
Suppose that ℓ ( c ) has the Fatou property. Then, the followingstatements are equivalent: (a) M defines a continuous linear operator M : ℓ → ℓ ( c ) . (b) C j ∈ ℓ ( c ) for all j and sup j k C j k ℓ ( c ) < ∞ .Proof. (a) ⇒ (b) Let K > k M x k ℓ ( c ) ≤ K k x k ℓ for all x ∈ ℓ . Forevery j we have that C j = M e j ∈ ℓ ( c ). Moreover,sup j k C j k ℓ ( c ) = sup j k M e j k ℓ ( c ) ≤ K sup j k e j k ℓ = K. (b) ⇒ (c) Since ℓ ( c ) has the Fatou property then ℓ ( c ) = ℓ ( c ) ′′ with equal norms.Let x ∈ ℓ . First note that for every i we have that X j | a ij x j | = X j (cid:0) | C j | , e i (cid:1) | x j | ≤ X j k C j k ℓ ( c ) k e i k ℓ ( c ) ′ | x j |≤ k e i k ℓ ( c ) ′ k x k ℓ sup j k C j k ℓ ( c ) and so M x ∈ ℓ . Given y ∈ ℓ ( c ) ′ it follows that (cid:0) | y | , | M x | (cid:1) = X i | y i | (cid:12)(cid:12)(cid:12) X j a ij x j (cid:12)(cid:12)(cid:12) ≤ X i X j | a ij x j y i | = X j | x j | X i | a ij y i | = X j | x j | (cid:0) | C j | , | y | (cid:1) ≤ X j | x j | k C j k ℓ ( c ) k y k ℓ ( c ) ′ ≤ k y k ℓ ( c ) ′ k x k ℓ sup j k C j k ℓ ( c ) . Then
M x ∈ ℓ ( c ) ′′ = ℓ ( c ) and k M x k ℓ ( c ) = sup y ∈ B ℓ ( c ) ′ (cid:0) | y | , | M x | (cid:1) ≤ k x k ℓ sup j k C j k ℓ ( c ) . (cid:3) In what follows assume that ℓ ( c ) has the Fatou property, C j ∈ ℓ ( c ) for all j andsup j k C j k ℓ ( c ) < ∞ . Then, M defines a continuous linear operator M : ℓ → ℓ ( c )and so, by Remark 7.1 we have that M can be extended as ℓ M / / i " " ℓ ( c ) ℓ ( m M ) I mM ; ; and ℓ ( m M ) is the largest σ -order continuous quasi-B.f.s. to which M can be ex-tended as a continuous operator. Remark . For every x ∈ ℓ ( m M ) it follows that I m M ( x ) = M x and so M definesa continuous linear operator M : ℓ ( m M ) → ℓ ( c ). Indeed, take 0 ≤ x = ( x n ) ∈ ℓ ( m M ) and x k = P kj =1 x j e j ∈ ℓ . Since x k ↑ x pointwise and ℓ ( m M ) is σ -ordercontinuous it follows that x k → x in ℓ ( m M ). Then, since M = I m M on ℓ , wehave that M x k = I m M ( x k ) → I m M ( x ) in ℓ ( c ) and so pointwise. Hence, the i -thcoordinate P kj =1 a ij x j of M x k converges to the i -th coordinate of I m M ( x ) and thus M x = I m M ( x ) ∈ ℓ ( c ). For a general x ∈ ℓ ( m M ), we only have to take the positiveand negative parts of x .From Proposition 7.2 applied to M : ℓ → ℓ ( c ) and Remark 7.8 we obtain thefollowing conclusion. Proposition 7.9.
The following statements are equivalent: (a) M defines a continuous linear operator M : ℓ p → ℓ ( c ) . (b) M is p -th power factorable with a continuous extension. (c) ℓ p ⊂ ℓ ( m M ) . (d) ℓ ⊂ ℓ p ( m M ) ∩ ℓ ( m M ) . (e) There exists
C > such that (cid:13)(cid:13)(cid:13) X j ∈ M x j C j (cid:13)(cid:13)(cid:13) ℓ ( c ) ≤ C (cid:16) X j ∈ M x pj (cid:17) p for all M ∈ P F ( N ) and ( x j ) j ∈ M ⊂ [0 , ∞ ) .Proof. The equivalence among statements (b), (c), (d), (e) is given by Proposition7.2. The stamement (a) implies (b) obviously. From Remark 7.8 we have that M defines a continuous linear operator M : ℓ ( m M ) → ℓ ( c ), so (c) implies (a). (cid:3) PTIMAL EXTENSIONS FOR p -TH POWER FACTORABLE OPERATORS 21 Let us give two conditions guaranteeing that M defines a continuous linear op-erator M : ℓ p → ℓ ( c ):(I) If p ′ is the conjugate exponent of p and P k C j k p ′ ℓ ( c ) < ∞ , then (e) in Proposi-tion 7.9 holds. Indeed, for every M ∈ P F ( N ) and ( x j ) j ∈ M ⊂ [0 , ∞ ) we havethat (cid:13)(cid:13)(cid:13) X j ∈ M x j C j (cid:13)(cid:13)(cid:13) ℓ ( c ) ≤ X j ∈ M x j k C j k ℓ ( c ) ≤ (cid:16) X j ∈ M x pj (cid:17) p (cid:16) X j ∈ M k C j k p ′ ℓ ( c ) (cid:17) p ′ ≤ (cid:16) X k C j k p ′ ℓ ( c ) (cid:17) p ′ (cid:16) X j ∈ M x pj (cid:17) p . (II) If M is p -power dominated, that is, there exists C > (cid:13)(cid:13)(cid:13) X j ∈ M x p j C j (cid:13)(cid:13)(cid:13) pℓ ( c ) ≤ C sup N ⊂ M (cid:13)(cid:13)(cid:13) X j ∈ N x j C j (cid:13)(cid:13)(cid:13) ℓ ( c ) for every M ∈ P F ( N ) and ( x j ) j ∈ M ∈ [0 , ∞ ), then (b) in Proposition 7.9 holdsby Lemma 7.5.For instance, in the case when ℓ ( c ) = ℓ q and a ij ≥ i, j , condition (II) issatisfied if F i ∈ ℓ for all i and P k F i k qℓ < ∞ , where F i denotes the i -th file of M .Indeed, for every M ∈ P F ( N ) and ( x j ) j ∈ M ∈ [0 , ∞ ), applying H¨older’s inequalitytwice for p and its conjugate exponent p ′ , we have that (cid:13)(cid:13)(cid:13) X j ∈ M x p j C j (cid:13)(cid:13)(cid:13) pℓ q = (cid:16) X i (cid:16) X j ∈ M x p j a ij (cid:17) q (cid:17) pq = (cid:16) X i (cid:16) X j ∈ M x p j a p ij a − p ij (cid:17) q (cid:17) pq ≤ (cid:16) X i (cid:16) X j ∈ M x j a ij (cid:17) qp (cid:16) X j ∈ M a ij (cid:17) qp ′ (cid:17) pq ≤ (cid:16) X i (cid:16) X j ∈ M x j a ij (cid:17) q (cid:17) q · (cid:16) X i (cid:16) X j ∈ M a ij (cid:17) q (cid:17) pqp ′ ≤ (cid:13)(cid:13)(cid:13) X j ∈ M x j C j (cid:13)(cid:13)(cid:13) ℓ q (cid:16) X i k F i k qℓ (cid:17) pqp ′ . Note that sup N ⊂ M (cid:13)(cid:13)(cid:13) P j ∈ N x j C j (cid:13)(cid:13)(cid:13) ℓ q = (cid:13)(cid:13)(cid:13) P j ∈ M x j C j (cid:13)(cid:13)(cid:13) ℓ q as a ij ≥ i, j . References [1] C. Bennett and R. Sharpley,
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